0.1 Upcrossing Inequality
Let \((X_{n}, \mathcal{F}_{n})\) be a submartingale, \(a<b\), and \(U_{n}(a,b;X)\) be the number of upcrossings of \([a,b]\) by \(X=(X_{n})\) up to time \(n\). Then \[ \mathbb{E}[U_{n}(a,b;X)]\leq \frac{1}{b-a} \mathbb{E}[(X_{n}-a)^+]. \]
\begin{proof}
\end{proof}